NBER WORKING PAPER SERIES A THEORY OF LIQUIDITY AND REGULATION OF FINANCIAL INTERMEDIATION. Emmanuel Farhi Mikhail Golosov Aleh Tsyvinski

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1 NBE WOKING PAPE SEIES A THEOY OF LIQUIDITY AND EGULATION OF FINANCIAL INTEMEDIATION Emmanuel Farhi Mikhail Golosov Aleh Tsyvinski Working Paper NATIONAL BUEAU OF ECONOMIC ESEACH 050 Massachusetts Avenue Cambridge, MA 0238 March 2007 Golosov and Tsyvinski acknowledge support by the National Science Foundation. We thank Daron Acemoglu, Stefania Albanesi, Franklin Allen, Marios Angeletos, icardo Caballero, V.V. Chari, Ed Green, Christian Hellwig, Skander Van den Heuvel, Oleg Itskhoki, Bengt Holmstrom, Guido Lorenzoni, Chris Phelan, Bernard Salanie, Jean Tirole, Ivan Werning, and uilin hou for comments. We also thank the audiences at the Bank of Canada, Minneapolis Federal eserve, Harvard, MIT, UT Austin, Wharton, and Society of Economic Dynamics for useful comments. The views expressed herein are those of the author(s) and do not necessarily reflect the views of the National Bureau of Economic esearch by Emmanuel Farhi, Mikhail Golosov, and Aleh Tsyvinski. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including notice, is given to the source.

2 A Theory of Liquidity and egulation of Financial Intermediation Emmanuel Farhi, Mikhail Golosov, and Aleh Tsyvinski NBE Working Paper No March 2007 JEL No. E6,G8,G2,G28 ABSTACT This paper studies a mechanism design model of financial intermediation. There are two informational frictions: agents receive unobservable shocks and can participate in markets by engaging in trades unobservable to intermediaries. Without regulations, intermediaries provide no risk sharing because of an externality arising from arbitrage opportunities. We identify a simple regulation -- a liquidity requirement -- that corrects such an externality by affecting the interest rate on the markets. We characterize the form of the optimal liquidity adequacy requirement for a general class of preferences. We show that whether markets underprovide or overprovide liquidity, and whether a liquidity cap or a liquidity floor should be used depends on the nature of the shocks that agents experience. Moreover, we prove that the optimal liquidity adequacy requirement implements a constrained efficient allocation subject to unobservable types and trades. We provide closed form solutions for the optimal liquidity requirement and welfare gains of imposing such requirements for two important special cases. In contrast with the existing literature, the necessity of regulation does not depend on exogenous incompleteness of markets for aggregate shocks. Emmanuel Farhi Department of Economics Harvard University Littauer Center 837 Cambridge St. Cambridge, MA 0238 efarhi@fas.harvard.edu Mikhail Golosov MIT Department of Economics E52-243G 50 Memorial Drive Cambridge, MA 0242 and NBE golosov@mit.edu Aleh Tsyvinski Department of Economics Harvard University Littauer Center 837 Cambridge St. Cambridge, MA 0238 and NBE tsyvinski@harvard.edu

3 A Theory of Liquidity and egulation of Financial Intermediation Emmanuel Farhi, Mikhail Golosov, and Aleh Tsyvinski February 23, 2007 Abstract This paper studies a mechanism design model of nancial intermediation. There are two informational frictions: agents receive unobservable shocks and can participate in markets by engaging in trades unobservable to intermediaries. Without regulations, intermediaries provide no risk sharing because of an externality arising from arbitrage opportunities. We identify a simple regulation a liquidity requirement that corrects such an externality by a ecting the interest rate on the markets. We characterize the form of the optimal liquidity adequacy requirement for a general class of preferences. We show that whether markets underprovide or overprovide liquidity, and whether a liquidity cap or a liquidity oor should be used depends on the nature of the shocks that agents experience. Moreover, we prove that the optimal liquidity adequacy requirement implements a constrained e cient allocation subject to unobservable types and trades. We provide closed form solutions for the optimal liquidity requirement and welfare gains of imposing such requirements for two important special cases. In contrast with the existing literature, the necessity of regulation does not depend on exogenous incompleteness of markets for aggregate shocks. Keywords: Optimal egulations, Financial Intermediation, Optimal Contracts, Market Failures, Mechanism Design. Introduction The role of nancial intermediaries in providing liquidity is one of the central features of a modern nancial system. Accordingly, the regulation of nancial intermediaries is an important function of central banks and is a topic of frequent debates in the policy-making community. In this paper we Farhi: Harvard University; Golosov: MIT and NBE; Tsyvinski: Harvard and NBE. Golosov and Tsyvinski acknowledge support by the National Science Foundation. We thank Daron Acemoglu, Stefania Albanesi, Franklin Allen, Marios Angeletos, icardo Caballero, V.V. Chari, Ed Green, Christian Hellwig, Skander Van den Heuvel, Oleg Itskhoki, Bengt Holmstrom, Guido Lorenzoni, Chris Phelan, Bernard Salanié, Jean Tirole, Ivan Werning, and uilin hou for comments. We also thank the audiences at the Bank of Canada, Minneapolis Federal eserve, Harvard, MIT, UT Austin, Wharton, and Society of Economic Dynamics for useful comments.

4 answer several important questions. Can markets provide the correct amount of liquidity? What is a precise nature of market failure if such exists? Can a regulator design a simple policy to improve on the allocations provided by competitive markets alone? The questions of public versus private provision of liquidity, limits of the markets in the provision of liquidity, and the role of government in regulation of nancial intermediation has been a subject of considerable volume of recent research. Most notably, Holmstrom and Tirole (998) and Allen and Gale (2004) study models of provision of liquidity in the presence of either informational or enforcement frictions. We study a mechanism design model of nancial intermediaries as providers of liquidity similar to Diamond and Dybvig (983) and Allen and Gale (2004). In our setup, agents receive unobservable taste shocks. Intermediaries invest in short and long term assets and o er a risk-sharing contract that pools risk across agents. An environment in which the only informational friction is unobservability of agents types is well analyzed in the literature. It is easy to show, as in Prescott and Townsend (984) and Allen and Gale (2004), that markets provide optimal allocations, and, therefore, there is no role for government intervention. The focus of this paper is on an environment in which there is an additional informational friction: consumers can trade assets unobservably on a private market by engaging in hidden side trades. Since the contribution of Jacklin (987), the possibility of agents engaging in hidden side trades has been recognized as an important constraint on the provision of liquidity by nancial intermediaries. 2 Unobservability of consumption arising from a possibility of such trades is a realistic and signi cant friction a ecting liquidity. It is di cult, if not impossible, for an individual nancial intermediary to control the exact use of funds or consumption of an agent or preclude a rm from engaging in bene cial trades with other rms in the economy. A di erent interpretation of unobservability of consumption is non-exclusivity of contracts. It is di cult for an individual nancial intermediary to preclude an agent to enter in additional risk sharing contracts with other intermediaries. Possibility of hidden trades can signi cantly worsen and even eliminate risk sharing. For example, Jacklin (987) and, more recently, Allen and Gale (2004) showed that absent any government regulations, in an environment with unobservable trades, intermediaries provide no risk sharing. Allen and Gale (2004) then conclude that, in the absence of aggregate shocks and incompleteness of the markets for aggregate risk, there is no regulation that can improve upon the market equilibrium. In contrast to the literature, we propose and analytically characterize a simple intervention that can improve upon the market allocations. We propose imposing a liquidity requirement that stipulates either the minimal (liquidity oor) or the maximal (liquidity cap) amount of liquidity holdings of the short asset for an intermediary. Such regulation a ects the interest rate on the hidden trade market, relaxes incentive compatibility constraints, and improves welfare. Im- Agents in our setup can be also thought of as as rms or entrepreneurs experiencing shocks to their productive opportunities. The interpretation of our environment as a model of rms makes the paper similar to the setup of liquidity provision in the productive sector by Holmstrom and Tirole (998). 2 The importance of access to credit markets as a constraint on the optimal program was also emphasized by Allen (985) and Chiappori, Macho, ey, and Salanie (994). 2

5 portantly, we also show that the optimal liquidity adequacy requirement implements the e cient allocation in which the social planner is constrained by the unobservability of agents types and the possibility of hidden trades. Therefore, the optimal liquidity adequacy requirement is not just the best requirement within a particular class of interventions, i.e. within a class of liquidity adequacy requirements. It is also an intervention that allows to achieve the highest possible welfare subject to these two information frictions. We identify a reason for the market failure an externality in which intermediaries do not internalize how liquidity they provide a ects other intermediaries via the possibility of trades on private markets. Importantly, this externality exists even when there are no aggregate shocks. This contrasts with the conclusions of Holmstrom and Tirole (998) and Allen and Gale (2004) that the government has a role in regulating liquidity only if there are aggregate shocks. A technical contributions of the paper is an analytical characterization of the optimal liquidity regulation in terms of easily interpretable wedges and determination of the form of the optimal regulation (liquidity cap or liquidity oor) for a general speci cation of preferences. We also provide a closed form solution for the optimal regulation in two cases: for a setup with logarithmic utility and for the environment studied by Diamond and Dybvig (983). We prove that the particular form of preference shocks one assumes is crucial to determine the direction of the optimal liquidity requirement. The intuition for why a liquidity requirement improves upon a competitive market allocation is that it a ects prices (interest rates in our case) on private markets. In short, a change in the interest rate a ects the deviating agent who simultaneously claims a di erent type and engages in hidden trades more than an agent who truthfully announces his type, therefore, relaxing the incentive compatibility constraints. The direction of the deviation depends on the speci cation of the preference shocks. In the case of the liquidity shocks, a deviating agent wants to save. A liquidity oor reduces the interest rate and makes borrowing less attractive. In the case of discount shocks, a deviating agent wants to borrow. A liquidity cap increases the interest rate and makes saving less attractive. Our model suggests practical implications for regulation of nancial intermediation. Various types of intermediaries or di erent regions in a country, depending on the primary nature of the shocks that the agents whom they serve experience, should have di erent forms of liquidity regulations. There are two appealing features of liquidity requirements that make it suitable for policy implementation. First, it is simple to implement as it speci es only how much liquidity intermediaries should hold in the rst versus second period. Intermediaries are then left to determine how they service their individual consumers without any additional government intervention. Second, it does not shut down private markets. ather, aggregate manipulation of liquidity endogenously changes the interest rate on these markets. We structure the paper to follow the discussion of equilibria and e ciency concepts progressing from less constrained to more constrained problems. The least restrictive program is an optimal allocation subject only to feasibility constraint SP in this setup, there is no private information. We then characterize a competitive equilibrium CE 2 and a constrained e cient allocation SP 2 3

6 in which the only informational friction is unobservable types. As we discussed above, the welfare achieved by CE 2 is equal to welfare achieved by SP 2. We then describe a competitive equilibrium CE 3 for the case when types are unobservable, and agents can also engage in hidden trades. We characterize the form of the optimal liquidity adequacy requirement depending on the speci cation of preferences. We then show that the optimal liquidity adequacy requirement implements the constrained e cient allocation SP 3 in which a planner is constrained by two informational frictions private types and hidden trades. Generically, welfare is higher in the solution to SP 3 than welfare achieved by competitive markets CE 3. For an important special case, considered in Diamond and Dybvig (983) and Jacklin (986), competitive equilibria without regulations imply no risk sharing and autarchic allocations while the optimal liquidity adequacy achieves welfare of the unconstrained problem SP (in fact, the solutions to all three programs SP, SP 2, and SP 3 coincide in this particular case). 3 2 elationship to the literature This paper builds on a large literature of risk sharing by in the presence of liquidity shocks (Diamond and Dybvig 983; Jacklin 987; Bhattacharya and Gale 987; Hellwig 994; Diamond 997; Von Thadden 999; Caballero and Krishnamurthy 2003; Allen and Gale 2003, ). More generally, our paper ts in the literature of optimal allocations with unobservable taste shocks following Atkeson and Lucas (992). Our paper uses the mechanism design framework of an important paper by Allen and Gale (2004) to analyze the model of intermediation in the presence of private markets. Our results in the model with private markets di er signi cantly from their work. The result of Allen and Gale (2004) that an equilibrium is ine cient relies on exogenously imposed incompleteness of markets for trades among intermediaries when there are aggregate shocks. In the absence of incomplete markets for aggregate shocks or in the absence of aggregate shocks, Allen and Gale (2004) conclude that there is no role for regulation of liquidity or any other regulatory intervention. We show that a liquidity requirement can improve upon the competitive equilibrium by eliminating above described externality even when there are complete markets for aggregate shocks or when there are no aggregate shocks. The mechanism of how liquidity requirements a ects interest rates on private markets and the characterization of the optimal liquidity adequacy requirement is new to the literature on the provision of liquidity by nancial intermediaries. Moreover, we provide a theoretical characterization of the optimal liquidity adequacy requirement for a general speci cation of shocks and closed form solutions for two important cases. Our paper shares a common goal with the work of Allen and Gale (2004) in studying whether laissez-faire markets provide too little or too much liquidity and whether a speci c policy interven- 3 We use terminology SP, SP 2, and SP 3 to correspond to what in the literature is, somewhat imprecisely, called rst-, second-, and third-best problems. The advantage of our notation is that we clearly de ne constraints that a social planner faces. 4 For a survey of the literature see Freixas and ochet (997) and Gorton and Winton (2002). 4

7 tion that occurs at an aggregate level can be Pareto improving or even optimal. Both of the papers direct regulations at intermediaries rather than individual consumers. A government regulates intermediaries while intermediaries on their own solve incentive problems via direct interactions with consumers. Holmstrom and Tirole (998) provide a theory of liquidity in a model in which intermediaries have borrowing frictions. Similar to our paper they do not assume incomplete markets. In their model, a government has an advantage over private markets as it can enforce repayments of borrowed funds while the private lenders cannot. They show that availability of government provided liquidity leads to a Pareto improvement when there is aggregate uncertainty. The role of the government in our model is to correct an ine ciency arising because of an externality associated with private information and possibility of hidden trades. In our paper, in contrast with Holmstrom and Tirole (998) and Allen and Gale (2004), a liquidity requirement improves upon a market allocation even when there is no aggregate uncertainty. Our paper also di ers conceptually from the seminal paper of Jacklin (987). That paper compares a competitive equilibrium with private markets CE 3 to the social optimum without private market SP 2 which is, essentially, equivalent to the statement that prohibition of private markets leads to a Pareto improvement. In our paper, we nd the optimal liquidity requirement and show that it implements the solution of the social planner s problem who is faced with both unobservable types and private markets SP 3 which, for this speci cation of preferences, coincides with SP and SP 2. In contrast with Jacklin, there is no need to prohibit private markets to achieve superior or even unconstrained allocations. A regulator can impose a liquidity adequacy requirement that achieves such optimal allocations. Lorenzoni (2006) considers a Diamond-Dybvig model of banking with nancial markets. His results on the characterization of the optimum is similar to our results for the special case of Diamond-Dybvig setup. The focus of Lorenzoni (2006) is on the models of money and on implementation of the optimum and advantages of various policy interventions. In his model, a speci cation of technology for intertemporal transfer of resources allows him to consider tradeo s of various policy interventions. Another paper that is related to our results in the Diamond-Dybvig setup is Caballero and Krishnamurthy (2003). They develop a model of an emerging market crisis in which there is a market for external borrowing and a domestic private market. The domestic market in their model is similar to the private market in our formulation. They show that the equilibrium coincides with the optimal allocation in the presence of private markets. They further show that a range of nancial instruments including liquidity requirements and taxes on external borrowing can implement the optimal allocation without private markets that coincides with the full information optimum. In our general model, a competitive equilibrium with the optimal liquidity adequacy requirement is di erent from the competitive equilibrium without private markets and, therefore, is di erent from unconstrained " rst-best" allocation. However, we show that in a special case of the Diamond and Dybvig (983) environment, the optimal liquidity regulation implements the unconstrained optimum. 5

8 While the focus of this paper is on the models of nancial intermediation, we also contribute to the literature on optimal taxation in the presence of hidden trades 5. In particular, Golosov and Tsyvinski (2006) study an optimal dynamic Mirrlees taxation with endogenous private markets. There are two main di erences between our paper and their work. The rst di erence is conceptual. In Golosov and Tsyvinski (2006) as in most of the models of dynamic Mirrlees taxation (see, e.g., Golosov, Kocherlakota, and Tsyvinski 2003 or a review in Golosov, Tsyvinski, and Werning 2006 and Kocherlakota 2006), private information (skill shocks) is dynamic and separable from consumption. The ine ciency of the competitive equilibrium in Golosov and Tsyvinski (2006) arises because of the dynamic nature of the private shocks. In our model, private information does not change stochastically over time, and the ine ciency arises due to non-separability of shocks and consumption. The second di erence is in the extent of the results that we obtain. Golosov and Tsyvinski (2006) and Bisin et. al. (200) are able to identify only the direction of a local policy change that leads to a Pareto improvement. We characterize the optimal allocation in the presence of private markets and show that the optimal liquidity regulation implements the optimum SP 3. We derive an analytical solution for the interest rate associated with the optimal liquidity requirement in terms of an easily interpretable wedge depending on the speci cation of preferences and distribution of shocks. Moreover, we provide a complete closed form solution for the optimum for two important examples. Studying optimal rather than locally improving interventions is not only interesting from the theoretical point of view. Optimal interventions may achieve a signi cant improvement in welfare compared to the competitive equilibrium. For example, we show that in the case of Diamond and Dybvig (983) the optimal liquidity requirement implements the unconstrained optimum. elated is a recent paper by Albanesi (2006) that studies a model of entrepreneurship and nancial assets. The focus of that paper is on an implementation of the optimal program with observable consumption as a competitive equilibrium with taxes in which agents can trade multiple assets. She derives a general result on di erential asset taxation in such models. In Diamond (997), as in our paper, the optimal allocation is di erent from autarky. result relies on the assumption that some consumers are exogenously restricted from participating in private markets. Unlike that paper, in our model all consumers can participate in markets. An elegant paper by Bisin and ampini (2004) justi es an institution of bankruptcy in a model of non-exclusive contracts. In their work, borrowers (entrepreneurs) have an access to secondary markets. A possibility of default on these secondary contracts worsens return to hidden borrowing and lending and yields a Pareto improvement. One justi cation for reserve requirements is found in the existence of deposit insurance. The rationale given is usually as follows: deposit insurance encourages risk taking behavior of intermediaries (see, e.g., Merton 977) which can be controlled by requiring intermediaries to hold adequate 5 See, for example, Arnott and Stiglitz (986), (990), Greenwald and Stiglitz (986), and Hammond (987). Several recent papers such as Geanakoplos and Polemarchakis (2004) and Bisin, et. al. (200) showed in very general settings that economies with asymmetric informations are ine cient and argued for Pareto-improving anonymous taxes. His 6

9 levels of liquidity. In this argument, existence of one potentially suboptimal policy, deposit insurance, justi es necessity of another policy - reserve requirements. Typically, this literature, with the exception of Hellman, Murdock and Stiglitz (998, 2000), does not consider optimal policy in the absence of deposit insurance. This literature also does not pose a friction or a market failure and does not nd an optimal policy that can be deposit insurance, reserve requirement, some combination of those, or maybe neither. Our results on the optimality of reserve requirements do not rely on the existence of any other exogenously given policy. 3 Model We consider a standard mechanism design model of nancial intermediation similar to Diamond and Dybvig (983) and closest to Allen and Gale (2004). The economy lasts three periods (t = 0; ; 2) and is populated by a continuum of ex-ante identical agents, or investors. There are two assets (technologies) in the model. The short asset is a storage technology that returns one unit of consumption good at t + for each unit invested at t. Investment in the long asset has to be done at t = 0 to yield ^ units of the consumption good at t = 2. Investors only value consumption at dates and 2 and receive a private idiosyncratic preference shock at the beginning of date : We denote the preference shock by 2 = [ L ; H ]. Investor s preferences are represented by a utility function u(c ; c 2 ; ), where c t denotes consumption at date t =, 2. The utility function u(; ) is assumed to be concave, increasing, and continuous for every type. We also assume a single Assumption > 0. In the paper we are primarily interested in studying three types of preferences: discount factor shocks, liquidity shocks, and valuation-neutral shocks. We also provide a complete characterization of the model for the fourth set of preference described in Diamond and Dybvig (983) and used in Jacklin (987). Example Discount factor shocks: u(c ; c 2 ; ) = ^u (c ) + ^u (c 2 ) : In this case, agents di er by how much they value second period consumption. The rst feature of these preferences is that a planner would like to allocate a relatively higher amount of second period consumption to an agent with a higher shock. The second feature of these preferences is also important to our results. Consider two agents who are allocated the same consumption c = c 2 = c. An agent with higher would receive a higher lifetime utility of consumption from such allocation. This feature of preferences creates an incentive for the planner to allocate, if possible, a higher present value of consumption to the agent with higher. Example 2 Liquidity shocks: u(c ; c 2 ; ) = ^u (c ) + ^u (c 2 ) : 7

10 In this case, low, i.e., a high liquidity shock, is a shock that makes consumption at date particularly valuable. Similar to the case of the discount shocks, the second feature of these preferences is that an agent with lower has a higher lifetime utility of consumption than an agent with lower. 6 Up to now, we have modelled liquidity and discount shocks as preferences shocks, or in other words, consumption opportunity shocks. We now discuss how we can think of the model as an environment in which rms or entrepreneurs face investment opportunity shocks. The outline of such an extension is as follows. Suppose there is a continuum of investors with identical CAA utilities: u(c ; c 2 ) = exp( c ) exp( c ) for some common discount factor : Agents are committed to nance a xed size normalized to investment opportunity paying out ~ ^ at date 2 and learn their type at date : It can be easily shown that this model is isomorphic to our model of discount factor shocks with = exp( ( ~ + )): ^ Imagine now that an entrepreneur is committed to nance a project with a known return ~q normalized to one at date 2: They learn the exact resources needed for this investment,, at ^ date : It is now easy to see that this model is isomorphic to our model of liquidity shocks with = exp ( + ~q^ ) : We can therefore interpret agents receiving taste shocks as investors receiving investment opportunity shocks. Example 3 Valuation-neutral shocks: Let ^u (c) = c and u(c ; c 2 ; ) = = + ( ) = ^u (c ) + ^ = + ( ) = ^ ^u (c 2 ) : () If ^u (c) = log (c), then u(c ; c 2 ; ) = ( ) ^u (c ) + ^u (c 2 ) : In this case, agents di er in how valuable their consumption is across periods, but the second feature of the preferences that we described above is absent here, and all agents value the lifetime consumption stream equivalently. Note that in the case of the log utility, there is no need to normalize preferences by ^, and valuation-neutral preferences do not depend on technology. 6 A natural question arises whether uility speci cation of liquidity shocks ^u (c) + ^u (c2) is a renormalization of the discount shocks ^u (c ) + ^u (c 2), and that by dividing utility in the case of discount shocks by we would arrive to the model with liquidity shocks. It is true that both of preferences have the same marginal rates of substitutions. However, the preferences are di erent in the direction of the levels of lifetime levels of utilities. In the case of liquidity shocks, it is low that gives an agent a higher lifetime value of consumption. In the case of dicount factor shocks, it is exactly the opposite high leads to high lifetime value of consumption. 8

11 Example 4 Diamond-Dybvig preferences. Let 2 f0; g and > > ^ : U(c ; c 2 ; ) = ( )u(c ) + u(c + c 2 ): In the case of Diamond-Dybvig preferences, agents are of two types: those who need to consume in the rst period, and those who are indi erent between consuming in the rst and the second period. We begin by assuming that there is no aggregate uncertainty. The timing of the events is as follows. At t = 0, all individuals are (ex-ante) identical. At t =, each consumer gets an i.i.d. draw of his type. The probability distribution of being an investor of type is denoted by F (). We assume that the law of large numbers holds, and that the cross-sectional distribution of types is the same as the probability distribution F. One can, therefore, interpret F () as the number of agents of type below. The realization of a consumer s type is private information. Each consumer has an endowment of e units of a consumption good at time t = 0, and no endowment at dates and 2. We denote byfc () ; c 2 ()g an allocation of consumption across consumers. An allocation is feasible if it satis es the feasibility constraint given by c () + c 2 () df () e: (2) ^ We do not impose a sequential service constraint so there are no bank runs in our model. We also restrict our attention to pure strategies. In what follows, we also consider symmetric equilibria. 4 Benchmark: equilibrium and constrained e cient allocation without private markets In this section, we de ne and characterize a competitive equilibrium for the economy without private markets for hidden trades. In this environment, agents are allocated with consumption allocations depending on their types. Agents cannot engage in any unobservable transaction, and their consumption is therefore observable. 4. De nition of equilibrium CE 2 Consider a market with a continuum of intermediaries. We assume throughout the paper that all activities at an intermediary level are observable. In period 0, before the realization of idiosyncratic shocks, consumers deposit their initial endowment with the intermediary. An intermediary agrees to provide a stream of consumption fc () ; c 2 ()g. These contracts are o ered competitively, and there is free entry for intermediaries. Therefore, consumers sign a contract with the intermediary that promises the highest ex-ante expected utility. We denote the equilibrium utility for a consumer by U. After the contract is signed, a consumer, given his type, chooses a reporting strategy 0 (), 9

12 and receives consumption c 0 j ; c 2 0 j. There are no private markets in which consumers can participate, and agents actual consumption is equal to the allocated consumption. We assume that intermediaries can trade bonds b among themselves. Without aggregate uncertainty the market for trades among intermediaries is very simple, and we describe it in this section as it is useful for later extensions to the case of aggregate uncertainty. We denote by q the price of a bond b in period t = that pays one unit of consumption good in period 2. All intermediaries take this price as given. They also pay dividends d ; d 2 to its owners. At t = 0, the intermediary invests x = c () df () in the short asset and y = ^ c2 () df () in the long asset. We consider a symmetric equilibrium. The maximization problem of the intermediary that faces intertemporal prices q and the reservation utility U is s.t. max d + qd 2 + qb b= ^ (3) c;d;y;b c () + c 2 () df () + d + d 2 = ^ + qb b= ^ e; (4) u (c () ; c 2 () ; ) df () U ; (5) u (c () ; c 2 () ; ) u c 0 ; c 2 0 ; for 8; 0. (6) In problem (3), an intermediary is maximizing pro ts subject to three constraints. Constraint (4) is a budget constraint that requires that payments to consumers (c () ; c 2 ()), payments of dividends (d ; d 2 ), and net payments on bonds are feasible. Constraint (5) states that the expected utility of an agent is higher than an equilibrium level of utility. Finally, constraint (6) is an incentive compatibility constraint that states that an agent receives higher utility from truthfully announcing his type rather than announcing any other type 0. In equilibrium, competition among intermediaries forces them to have zero pro ts. We now de ne a competitive equilibrium. De nition A competitive equilibrium CE 2 is a set of allocations fc () ; c 2 ()g; a price q, dividends fd ; d 2 g ; bond trades b, and a utility level U such that (i) intermediaries choose ffc () ; c 2 ()g; fd ; d 2 g ; bg to solve problem (3) taking q and U as given; (ii) consumers choose a contract that o ers them the highest ex-ante utility; (iii) the aggregate feasibility constraint (2) holds; (iv) rms make zero pro ts; (v) bonds markets clear, b = 0. It is easy to show that, in equilibrium, =q = ^ and d = d 2 = 0: 0

13 4.2 Characterization and relationship to an unconstrained problem SP and to a constrained e cient problem SP 2 Intermediaries operate on competitive markets and, therefore, maximize an ex ante expected utility of agents. We can immediately see that the problem of the intermediary (3) in a competitive equilibrium is dual to the problem: max c ;c 2 s.t. feasibility (2) and incentive compatibility (6) hold. u (c () ; c 2 () ; ) df () ; (7) Problem (7) is a de nition of a particular notion of constrained e ciency which we denote by SP 2. In this problem, a planner receives reports from the agents of their types and provides a menu of consumption allocations fc () ; c 2 ()g to maximize expected utility of an agent subject to incentive compatibility and feasibility constraints. planner faces is unobservability of consumer types. The only informational friction that this The above result that constrained e cient allocations coincide with the competitive equilibrium does not mean that there is perfect risk sharing as the allocations in the problem SP 2 have to satisfy the incentive compatibility constraints. It is useful to de ne an unconstrained problem SP in which types of agents are observable. In that program the social planner does not face any constraints except for feasibility, i.e., maximizes the objective function in the problem (7) subject to (2). Obviously, the problem of SP is weakly less restrictive than problem SP 2. Therefore, the welfare achieved in SP is weakly higher than welfare achieved in SP 2. We summarize characterization of the solution to the problem (7) and, therefore, the solution of the problem (3) in the proposition that follows. The result is similar to Prescott and Townsend (984) and Allen and Gale (2004). Proposition (Optimum and competitive equilibrium with observable consumption) Let c () and c 2 () be equilibrium allocations in De nition. Then competitive equilibrium CE 2 is constrained e cient, i.e., solves problem SP 2. Moreover, 8 2 :. If preferences are discount shocks, as in the example ; bu 0 (c ()) ^bu 0 (c 2 ()); 2. If preferences are liquidity shocks, as in the example 2; bu 0 (c ()) ^bu 0 (c 2 ()); 3. If preferences are valuation-neutral shocks, as in the example 3, bu 0 (c ()) = ^ ( ) bu 0 (c 2 ()); Proof. In the appendix. The intuition for the wedge in parts and 2 of the above proposition is as follows. Consider, for example, the case of liquidity shocks. An agent with a low liquidity shock, high, has an incentive to report a high liquidity shock, low, to receive a higher consumption in period. A wedge (implicit tax) between the rst and the second period consumption relaxes the incentive constraint by making such deviations more costly. We can contrast the result above with the case

14 where the shocks are public information, the solution to the problem SP. In that case, there is no intertemporal wedge, and the Euler equation holds with equality. The case of the valuationpreferences in Part 3 of the proposition is special as the incentive compatibility constraint (6) does not bind. For such preferences, the solution to the problem SP coincides with the solution to the problem SP 2, and there is no wedge in the intertemporal valuation for all types. 7 5 Competitive equilibrium with private markets The allocations described in the previous section may not be achieved if agents can engage in transactions on markets. Allen (985) and Jacklin (987) were the rst to point out that the possibility of such trades may restrict or even lead to a complete elimination of risk sharing. In this section we de ne and characterize a competitive equilibrium allocation in the presence of private markets. 8 We rst argue, as in Jacklin (987) and Allen and Gale (2004), that without regulations nancial intermediaries provide no risk sharing. Unlike the previous literature we then identify a precise reason for the absence of insurance, an externality, that, as we show in the sections that follow, can be corrected by liquidity regulations. 5. Private market We model unobservability of consumption using the setup of private markets as follows. Consider an environment in which all consumers have access to a market in which they can trade assets among themselves unobservably 9. Formally, suppose that consumers are o ered a menu of contracts fc () ; c 2 ()g. We model private markets as an endowment economy where endowments are allocations that agents receive (possibly by misrepresenting their types). A consumer treats the contract and the equilibrium interest rate on the private market as given and, given his type, chooses his optimal reporting strategy 0 that determines his endowment of consumption c 0 ; c 2 0. Unlike in the environment without private markets, actual after-trade consumption fx () ; x 2 ()g may di er from the consumption speci ed in the contract, since it is impossible to preclude a consumer from borrowing and lending the amount s () on the private market. It can be easily shown that a consumer trades only a risk free security s () and solves: 7 Note that not every utility function of the form ^u (c )+( ) ^u (c 2) would imply that the incentive compatibility does not bind. For example, if we did not normalize by ^ and in our de niton of value neutral preferences (), and instead had u (c ; c 2; ) = c = ( ) + ( ) c 2 = ( ) ; the solution of the problem SP 2 would feature a wedge in the intertemporal valuation, and the incentive compatibility constraint would bind. 8 An alternative interpretation of the assumption of the private markets is non-exclusivity by which we mean that it is impossible for an intermediary to observe or control transactions of a consumer with other intermediaries. 9 All our analysis is easily extended to the case in which agents can trade not only among themselves but also with other intermediaries. This case would bring this model closer to an interpretation as an environment of non-exclusive contracts. Key assumption that allows us to extend our results to that case is that portfolios of the intermediary (investment in short and long assets) are observable while transactions with individual consumers are not observable. Our choice of modelling side trades as private markets allows us to economize on notation without a ecting the substance of the results. 2

15 max u (x x ();x 2 ();s(); 0 () ; x 2 () ; ) ; (8) s.t. 8i: x () + s () = c 0 ; (9) x 2 = c s () : (0) We denote the value of this problem by V ~ (fc () ; c 2 ()g ; ; ). We denote by 0 () the reporting strategy that the agent chooses in the problem above. An equilibrium in the private market requires that in each period the total endowment of consumption goods be equal to the total after trade consumption for t = ; 2: x t 0 () df () = c t 0 () df () : () Conditional on reports, the private market economy is a standard endowment economy. Therefore, aggregate endowments c 0 df () and c 2 0 df () determine the interest rate. We now de ne equilibrium in the private market. De nition 2 An equilibrium in the private market given the pro le of contracts fc () ; c 2 ()g consists of an interest rate ; and, for each agent : strategies 0 (), and allocations fx () ; x 2 () ; s ()g such that (i) consumers solve problem (8) taking ffc () ; c 2 ()g ; g as given; (ii) the feasibility constraints on the private market () are satis ed. We assume that for any menu of contract fc () ; c 2 ()g that is o ered there exists a unique equilibrium. 5.2 Competitive equilibrium with private markets CE 3 In the presence of private markets, intermediaries need to take into account, in addition to unobservable types, that consumers are able to engage in transactions in the private market. Each intermediary chooses payments fc () ; c 2 ()g; pays dividends d ; d 2, and trades bonds b with other intermediaries. It is important to note that intermediaries take the interest rate on the private market as given. The maximization problem of the intermediary that faces an intertemporal price q, a price on the private market, and a reservation utility U is max d + qd 2 + qb b= ^ (2) c;d;y;b s.t. c () + c 2 () df () + d + d 2 = ^ ^ + qb b= ^ e; (3) ~V (fc () ; c 2 ()g ; ; ) ~ V ( c 0 ; c 2 0 ; ; ); 8; 0 ; (4) 3

16 ~V (fc () ; c 2 ()g ; ; )df () U : (5) The rst constraint in the intermediary s problem is the budget constraint. The second constraint is incentive compatibility that states that, given interest rates, consumers choose to truthfully reveal their types. The last constraint states that the intermediary cannot o er a contract which delivers a lower expected utility than the equilibrium utility U from the contracts o ered by other intermediaries. In equilibrium, all intermediaries act identically and make zero pro ts. The intermediary s problem in this economy is very similar to that in the economy with observable trades. The only di erence comes from the fact that the incentive constraint (4) takes into account side trades that are not observable. The de nition of the competitive equilibrium is parallel to that in the economy with observable trades. De nition 3 A competitive equilibrium CE 3 is a set of allocations fc () ; c 2 ()g; a price q, dividends fd ; d 2 g ; bond trades b, utility U, and the interest rate on the private market such that (i) intermediaries choose ffc () ; c 2 ()g; fd ; d 2 g ; bg to solve problem (2) taking q; ; and U as given; (ii) consumers choose the contract that o ers them the highest ex-ante utility; (iii) the aggregate feasibility constraint (2) holds; (iv) the private market, given the menus fc () ; c 2 ()g, is in equilibrium, and is an equilibrium price; (v) rms make zero pro ts; (vi) bonds markets clear, b = 0. First, we show a straightforward lemma that the incentive compatibility constraint (4) takes the form of equalizing present value of intertemporal allocations across periods because, otherwise, an agent would pretend to claim a type that gives a higher present value of allocations and engage in trades on the private markets to achieve desired consumption allocations. Lemma An allocation of consumptions satis es incentive compatibility constraint (4) i c () + c 2 () = c 0 + c 2 0 for any, 0. (6) Let us rewrite the problem of the intermediary in a more manageable form by considering its dual, simplifying incentive compatibility constraint using Lemma, and using the fact that d = d 2 = b = 0 to reduce to: max c ;c 2 u (c () ; c 2 () ; ) df () ; (7) s.t. (6) and c () + c 2 () df () e: (8) ^ 4

17 It is easy to see that the interest rates on the markets for trades among intermediaries must be equal to the return on the production technology, so that =q = ^. We now argue that = ^; otherwise, arbitrage opportunities are created. For example, suppose that < ^, i.e., an interest rate on the private market is lower than. An intermediary then chooses to invest only in the long asset and sets c 2 () ^ df () = e and c () df () = 0. Consumers then can borrow on the private market at the interest rate that is lower than the technological rate of return ^ available to the intermediary. Therefore, the only price that can be an equilibrium price is = ^ so that intermediaries do not engage in arbitrage. We can summarize this reasoning in the following proposition. Proposition 2 (Absence of risk-sharing without regulations) Let denote equilibrium price on the private market corresponding to the competitive equilibrium in De nition 3. Then = ^: The only allocation that competitive markets can achieve in such an economy is an autarcic allocation in which the present values of endowments evaluated at ^ are equated across di erent types: c () + c 2 () ^ = c 0 + c 2 0 ^ for any, 0. This proposition implies that there is complete absence of risk sharing as in Jacklin (987) and Allen and Gale (2004). There are two important concepts that this proposition summarizes. The rst is that the incentive compatibility constraints (arising because of unobservability of types and possibility of trades) lead to equalization of present values of consumptions evaluated at the interest rate on the private market. The second fact is that arbitrage among competitive intermediaries forces the equilibrium interest rate on the private market to be equal to the return on savings ^. Intuitively, the reason that the competitive equilibrium achieves only an autarcic allocation is an externality. Intermediaries do not take into account how the contracts o ered to its investors a ect the return on trades and thus incentives to reveal information truthfully for consumers of other intermediaries. Individual intermediaries can not internalize this e ect. Competition between di erent intermediaries implies that interest rates at which consumers trade are equated to ^. The interpretation of our result is di erent from Jacklin (987) and Allen and Gale (2004) as we describe it as an externality that we show in the next section can be corrected by a government intervention. 6 Optimal liquidity requirements In this section we show that there exists an intervention a liquidity requirement imposed on intermediaries that improves upon the competitive equilibrium allocation. We then determine the best (optimal) liquidity requirement. We show how the form of the optimal liquidity adequacy depends on the nature of the shocks that agents experience. Finally, we show that the best liquidity requirement implements a particular notion of constrained e cient allocations. The key concept in this section is that a manipulation of liquidity leading to changes in the interest rate on private 5

18 markets, may lead to an improvement of risk sharing even in the presence of trading possibilities by agents contrasting with the results of Jacklin (987) and Allen and Gale (2004). 6. De nition and e ects of a liquidity requirement A liquidity requirement is a constraint imposed on all intermediaries, i.e., a constraint on the problem (2) that requires that investment in the short asset for any intermediary should be higher (lower) than a level i c () df () i: (9) We call a liquidity requirement a liquidity cap if (9) is imposed with less or equal sign. A liquidity cap stipulates the maximal amount of the short asset that an intermediary can hold. We call a liquidity requirement a liquidity oor if (9) is imposed with a greater or equal sign. A liquidity oor stipulates the minimal amount of the short asset that an intermediary can hold. An attractive feature of the liquidity requirement is that it does not require a regulator to observe individual contracts c () only an aggregate portfolio allocation of the intermediaries needs to be observed. We now intuitively describe the e ects that a binding liquidity requirement has on the interest rate on private markets. Let ^c () be the allocation of consumption that arises in an equilibrium without government intervention in De nition 3. Suppose that a liquidity oor i is set higher than the amount of aggregate liquidity provided by competitive markets: i > ^{; where ^{ = ^c () df (). When a liquidity oor is imposed, the aggregate endowment in the private markets in the rst period c () df () is equal to i rather than ^{. ecall that private trading markets in which agents participate after receiving their allocation from the intermediaries are an endowment economy. The liquidity oor increases the rst period aggregate endowment in the private market (and, correspondingly, decreases the second period endowment) and, therefore, lowers the interest rate such that < ^. Imposing a binding liquidity cap has the opposite e ect as it lowers the rst period aggregate endowment and, therefore, increases such that > ^. The mechanism by which a liquidity requirement a ects the interest rate on the private markets is a key to understanding the main idea behind how our model works. In the absence of regulations, it is impossible for the interest rate on the private market to di er from ^. As we showed in Proposition 2, an intermediary would engage in arbitrage and would not internalize possible adverse e ects that such arbitrage has on the provision of incentives and risk-sharing in the economy. The liquidity requirement puts a limit on the minimal (maximal) rst period payments (liquidity) an intermediary can make and limits arbitrage by intermediaries. Why may it be the case that, for example, a decrease in the private market interest rate (that 6

19 corresponds to a binding liquidity oor i > ^{) improves welfare and risk sharing? There are two e ects of decreasing the private market interest rate. First, it is clear that a decrease in below ^ decreases welfare as it introduces an intertemporal wedge in the marginal utilities of agents. There is also a second e ect. ecall that unobservability of agents types and possibility of trades require that agents of various types receive the same present value of consumption evaluated at the private market interest rate : c () + c 2 () = c 0 + c 2 0 : For a given level of the present value of consumption, a regulator has a policy instrument changing the interest rate. Therefore, the amount of resources evaluated at the real rate of return may di er across agents c () + c 2 () ^ 6= c 0 + c 2 0 ^ : A change in the interest rate leads to a relative redistribution of resources from the rst to the second period which may bene t an agent who derives a higher utility from a given present value of consumption streams and lead to an improvement in the ex-ante welfare. Competitive markets lack this additional instrument because of arbitrage and the fact that each individual rm cannot set the interest rate. A regulator, however, can a ect the interest rate and achieve allocations better than autarcic allocations achieved by the markets. We show next how the exact form of the liquidity requirement and a corresponding direction of the private market interest rate change depend on the form of agents preferences. 6.2 Optimal liquidity regulations We rst simplify the problem of characterizing an equilibrium with a liquidity adequacy requirement. Let subject to V (I; ; ) = max x ;x 2 u(x ; x 2 ; ) (20) x + x 2 I, (2) be the ex-post indirect utility of an investor of type if her income is I, and the interest rate on the private market is. Denote by x u (I; ; ) and xu 2 (I; ; ) the uncompensated demand functions in this problem. It is easy to see that the problem of nding an optimal liquidity requirement is to choose the interest rate and income I to maximize the expected indirect utility of agents subject to feasibility constraints. subject to max I; V (I; ; )df () (22) x u (I; ; ) + xu 2 (I; ; ) df () e; (23) ^ 7